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Mirrors > Home > MPE Home > Th. List > oduleval | Structured version Visualization version GIF version |
Description: Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduval.d | ⊢ 𝐷 = (ODual‘𝑂) |
oduval.l | ⊢ ≤ = (le‘𝑂) |
Ref | Expression |
---|---|
oduleval | ⊢ ◡ ≤ = (le‘𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6424 | . . . . 5 ⊢ (le‘𝑂) ∈ V | |
2 | 1 | cnvex 7348 | . . . 4 ⊢ ◡(le‘𝑂) ∈ V |
3 | pleid 16369 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
4 | 3 | setsid 16239 | . . . 4 ⊢ ((𝑂 ∈ V ∧ ◡(le‘𝑂) ∈ V) → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
5 | 2, 4 | mpan2 683 | . . 3 ⊢ (𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
6 | 3 | str0 16236 | . . . 4 ⊢ ∅ = (le‘∅) |
7 | fvprc 6404 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (le‘𝑂) = ∅) | |
8 | 7 | cnveqd 5501 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ◡∅) |
9 | cnv0 5753 | . . . . 5 ⊢ ◡∅ = ∅ | |
10 | 8, 9 | syl6eq 2849 | . . . 4 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ∅) |
11 | reldmsets 16212 | . . . . . 6 ⊢ Rel dom sSet | |
12 | 11 | ovprc1 6916 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
13 | 12 | fveq2d 6415 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) = (le‘∅)) |
14 | 6, 10, 13 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
15 | 5, 14 | pm2.61i 177 | . 2 ⊢ ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
16 | oduval.l | . . 3 ⊢ ≤ = (le‘𝑂) | |
17 | 16 | cnveqi 5500 | . 2 ⊢ ◡ ≤ = ◡(le‘𝑂) |
18 | oduval.d | . . . 4 ⊢ 𝐷 = (ODual‘𝑂) | |
19 | eqid 2799 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
20 | 18, 19 | oduval 17445 | . . 3 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
21 | 20 | fveq2i 6414 | . 2 ⊢ (le‘𝐷) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
22 | 15, 17, 21 | 3eqtr4i 2831 | 1 ⊢ ◡ ≤ = (le‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ∅c0 4115 〈cop 4374 ◡ccnv 5311 ‘cfv 6101 (class class class)co 6878 ndxcnx 16181 sSet csts 16182 lecple 16274 ODualcodu 17443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-dec 11784 df-ndx 16187 df-slot 16188 df-sets 16191 df-ple 16287 df-odu 17444 |
This theorem is referenced by: oduleg 17447 odupos 17450 oduposb 17451 oduglb 17454 odulub 17456 posglbd 17465 oduprs 30172 odutos 30179 ordtcnvNEW 30482 ordtrest2NEW 30485 |
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